This paper proposes a method to estimate state of charge (SoC) for Lithiumion battery pack (LIB) with 𝑁 series-connected cells. The cell’s model is represented by a second-order equivalent circuit model taking into account the measurement disturbances and the current sensor bias. By using two sigma point Kalman filters (SPKF), the SoC of cells in the pack is calculated by the sum of the pack’s average SoC estimated by the first SPKF and SoC differences estimated by the second SPKF. The advantage of this method is the SoC estimation algorithm performed only two times instead of 𝑁 times in each sampling time interval, so the computational burden is reduced. The test of the proposed SoC estimation algorithm for 7 samsung ICR18650 Lithium-ion battery cells connected in series is implemented in the continuous charge and discharge scenario in one hour time. The estimated SoCs of the cells in the pack are quite accurate, the 3-sigma criterion of estimated SoC error distributions is 0.5%.

1. International Journal of Electrical and Computer Engineering (IJECE)
Vol. 12, No. 2, April 2022, pp. 1334~1349
ISSN: 2088-8708, DOI: 10.11591/ijece.v12i2.pp1334-1349  1334
Journal homepage: http://ijece.iaescore.com
State of charge estimation for lithium-ion batteries connected in
series using two sigma point Kalman filters
Chi Nguyen Van1
, Thuy Nguyen Vinh2
1
Institute of High-Technology Research and Development for Industry (RIAT), Thai Nguyen University of Technology, Thai Nguyen,
Vietnam
2
Department of Automation, Thai Nguyen University of Technology (TNUT), Thai Nguyen, Vietnam
Article Info ABSTRACT
Article history:
Received Apr 19, 2021
Revised Aug 3, 2021
Accepted Sep 1, 2021
This paper proposes a method to estimate state of charge (SoC) for Lithium-
ion battery pack (LIB) with 𝑁 series-connected cells. The cell’s model is
represented by a second-order equivalent circuit model taking into account
the measurement disturbances and the current sensor bias. By using two
sigma point Kalman filters (SPKF), the SoC of cells in the pack is calculated
by the sum of the pack’s average SoC estimated by the first SPKF and SoC
differences estimated by the second SPKF. The advantage of this method is
the SoC estimation algorithm performed only two times instead of 𝑁 times
in each sampling time interval, so the computational burden is reduced. The
test of the proposed SoC estimation algorithm for 7 samsung ICR18650
Lithium-ion battery cells connected in series is implemented in the
continuous charge and discharge scenario in one hour time. The estimated
SoCs of the cells in the pack are quite accurate, the 3-sigma criterion of
estimated SoC error distributions is 0.5%.
Keywords:
Battery pack
Lithium-ion battery
Sigma point Kalman filter
SoC estimation
State of charge
This is an open access article under the CC BY-SA license.
Corresponding Author:
Chi Nguyen Van
Institute of High-Technology Research and Development for Industry (RIAT), Thai Nguyen University of
Technology National
666, 3/2 Street, Tich Luong Ward, Thai Nguyen City 251750, Vietnam
Email: ngchi@tnut.edu.vn
1. INTRODUCTION
Lithium-ion batteries pack (LIB) are so far known to be the most commonly used batteries with the
following advantages: a higher energy density, voltage capacity; lower self-discharge rate than other
rechargeable batteries; the price is much cheaper than the LiPo battery. LIBs power almost every portable
devices such as smartphone, laptop, and tablet sold today across the world. Now, LIBs are applied to not only
electric cars, electric motorcycles, buses, trucks but also large-scale electrical energy storage. LIB is formed
by many cells connected in series and parallel in order to get the desired voltage and power [1] as shown in
Figure 1. The largest market for LIBs has traditionally been portable electronic devices, but in the near future
there is an extensive growth in the demand for LIBs in transportation and electrical energy storages [2]. The
state of charge (SoC) is an important input for balancing, energy and power calculations of LIB. SoC
depends on the open-circuit-voltage (OCV) with the nonlinear relationship affected by the temperature,
aging, and charge/discharge mode, [3]–[5]. The OCV depends on temperature and electrode particle surface
concentration, but SoC depend on particle average concentration. In the practice, optimal safe operation
range of SoC for cells is above 20% and below 90%. Figure 2 describes a typical relationship between SoC
and OCV at -5 0
C and 25 0
C of temperature for a Lithium samsung cell. SoC is a parameter that is not
measurable directly, so it is estimated by using the current, voltage, and temperature of the cell. SoC is
needed to be estimated accurately in order to use for the SoC balance control problem. Controlling the

2. Int J Elec & Comp Eng ISSN: 2088-8708 
State of charge estimation for lithium-ion batteries connected in series using … (Chi Nguyen Van)
1335
operation of the battery system in order to prolong the life of the battery cells, to operate safely is a topic that
scientists are currently interested in applying LIBs in transportation and electrical energy storage. To estimate
SoC for one cell, we need the model describing the dynamic behavior of the cell and the SoC estimation
algorithm based on that model with real-time current, voltage, and temperature inputs. The accuracy of the
estimated SoC depends on the model of the cell and the SoC estimation algorithms used [6].
The model of the cell: if the complex model is used (such as an electrochemical model), the
estimated SoCs will be more accurate, but the computational burden and hardware costs will increase a lot
when the number of cells in series 𝑁 is large. If the equivalent-circuit model is used, the order of that model
plays an important role. The computational complexity of the SoC estimation algorithm is simpler with the
lower-order model but the cell dynamic behavior is not reflected accurately. There is higher accuracy in SoC
estimation when the higher-order model is used, but the computational burden is increased [7]. The SoC
estimation algorithm: Up to now, most of the SoC estimation algorithms are based on coulomb counting [8],
Kalman filter (KF) [9], adaptive extended Kalman filter (EKF) [10], [11], and other [12], [13] improved
algorithms. Several other algorithms such as the use of fuzzy models and neural networks are also applied
[14], [15].
Figure 1. Inside the tesla model S battery pack
(a) (b)
Figure 2. The typical relationship between SoC and OCV for a lithium samsung cell: (a) tested at -5 0
C and
(b) tested at 25 0
C
The SoC estimation for 𝑁 cell connected in series is the challenge problem, there are some works
related to this in literature [16], [17]. Normally, the SoC estimation algorithm is implemented repeatedly 𝑁
times, if the number of cells 𝑁 is large then this is really a computational burden for the battery management
system [18]. Beside that, from the works [7], [12], [14], [17] and the actual testing we found that the
interference of the measuring noises, current sensor bias are an obstacle to estimating SoC exactly. In
practice, the current through the cell is usually measured by a shunt resistor with a voltage signal of 𝑚𝑉, this
voltage signal is more susceptible to interference than the voltage measuring signal at both ends of the cell
(2.7 𝑉 to 4.2 𝑉). The current sensor bias of the rectifier affected by temperature needs to be considered in the

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Int J Elec & Comp Eng, Vol. 12, No. 2, April 2022: 1334-1349
1336
current measurement. The KF theory assumes that all noises are zero mean, so an unknown current sensor
bias can introduce permanent SoC error. This paper introduces the SoC estimation method for a LiB
connected in series using sigma point Kalman filters (SPKF) to minimize the influence of current and
vfoltage measurement noise on the estimated SoC results. By using the second-order equivalent circuit model
(SECM), this allows to describe more exactly the cell’s dynamic in the charged and discharged scenarios
with amplitude varying suddenly (an important feature of electrical vehicles using LIBs must be considered)
[19]. This method uses two SPKFs, the SoC of cells in the pack is calculated by sum of pack’s average SoC
estimated by first SPKF and SoC differences estimated by second SPKF.
The contributions of this work are: The first, by using only two filters to estimate the 𝑁 SoCs of 𝑁
cells connected in series, the SoC estimation algorithm is performed only two times instead of 𝑁 times in
each sampling time interval, so the computational burden reduced to 𝑁 − 2 times, this is a very effective
solution when the battery pack has a large number of cells. The second, the SoC estimation taking into
account the current sensor bias in the model in association with applying the SPKF filter allows mapping the
distributions of current, voltage, and temperature signals to the distribution of the estimated SoCs of the cells
with more accuracy tested by the 3 criterion. The remaining part of this paper is structured as follows:
section 2 mentions the SECM of the cell taking into account the measurement noises and the current sensor
bias. Section 3 introduces the proposed SoC estimation algorithm of cells connected in series using 02
SPKFs. Section 4 shows the results applied on the Lithium ICR18650-22P SAMSUNG cells and discussions.
Finally, section 5 summarizes some conclusions.
2. THE SECOND-ORDER EQUIVALENT CIRCUIT MODEL OF CELL
Suppose that the LIB’s cell is depicted as in Figure 3(a), in which 𝑖(𝑡) is the cell current (𝑖(𝑡) > 0
for charge and 𝑖(𝑡) < 0 for discharge, 𝑖(𝑡) = 0 for cell in no charge/discharge), 10% < 𝑧(𝑡) < 95% is SoC,
3.6𝑉 < 𝑣(𝑡) < 4.2𝑉 is the cell voltage, −50
𝐶 < 𝑇(𝑡) < 500
𝐶 is cell temperature. In order to estimate the
SoC at the sample time 𝑘, which is 𝑧𝑘, we need to measure the voltage, current and cell temperature. Let
𝑖𝑘 + 𝑏𝑘 + 𝑛𝑖,𝑘 is the measured current, 𝑣𝑘 + 𝑛𝑣,𝑘 is measured voltage, in which 𝑏𝑘 is the current sensor bias;
𝑛𝑖,𝑘 is the current noise; 𝑛𝑣,𝑘 is the voltage noise. The Figures 3(b), 3(c) are the recorded voltage and current
in the practice, respectively. The scanning electrochemical microscopy (SECM) of the cell is plotted in
Figure 4 with 𝑂𝐶𝑉(𝑧(𝑡), 𝑇) is the relationship between the oral cholera vaccine (OCV) and SoC, this
relationship is nonlinear function depending on cell temperature and this function is needed to determine by
the practice data collected by tests. The parameter ℎ(𝑡) is the dynamic hysteresis voltage as a function of SoC
and time, 𝑅1(𝑇), 𝐶1(𝑇), 𝑅2(𝑇), 𝐶2(𝑇) are two resistor-capacitor pairs describe the cell dynamic, 𝑅0(𝑇) is
series resistance parameter, 𝑀0(𝑇), 𝑀(𝑇) are the hysteresis parameters, 𝜂(𝑇) is Coulombic efficiency, 𝛾(𝑇)
is the parameter related to the the rate of decay. Note that all parameters of cell depend on the cell
temperature. In exact terms, they also depend on the cell aging, the charge/discharge condition. For different
kind of cell (Samsung, Panasonic), they are different. In this work, we suppose that all parameters of cell
only depend on the cell temperature.
Suppose that all the quantities is discreterized at the sampling time 𝑘, 𝑘 = 0,1,2, … , ∞ with the
sampling interval ∆𝑡 is small enough so that the current and voltage of cell are constant over that interval.
Suppose that 𝑄 is total capacity of a cell, the change in SoC over sampling interval is (1).
𝑧𝑘+1 = 𝑧𝑘 − 𝑖𝑘
𝜂𝑘Δ𝑡
𝑄
(1)
The diffused resistor currents are computed as (2),
𝑖𝑅1,𝑘+1 = 𝑒𝑥𝑝 (−
∆𝑡
𝑅1𝐶1
) 𝑖𝑅1,𝑘 + (1 − 𝑒𝑥𝑝 (−
∆𝑡
𝑅1𝐶1
)) 𝑖𝑘
𝑖𝑅2,𝑘+1 = 𝑒𝑥𝑝 (−
∆𝑡
𝑅2𝐶2
) 𝑖𝑅2,𝑘 + (1 − 𝑒𝑥𝑝 (−
∆𝑡
𝑅2𝐶2
)) 𝑖𝑘 (2)
Hysteresis voltage is given by (3).
ℎ𝑘+1 = 𝑒𝑥𝑝 (− |
𝑖𝑘𝜂𝑘∆𝑡
𝑄
|)ℎ𝑘 − (1 − 𝑒𝑥𝑝 (|
𝑖𝑘𝜂𝑘∆𝑡
𝑄
|))𝑠𝑖𝑔𝑛(𝑖𝑘) (3)

4. Int J Elec & Comp Eng ISSN: 2088-8708 
State of charge estimation for lithium-ion batteries connected in series using … (Chi Nguyen Van)
1337
(a)
(b) (c)
Figure 3. Illustration of the cell measurements, (a) the LIB, (b) the recorded voltage (c) current of cell
Figure 4. The second-order equivalent circuit model of the cell
The current sensor bias is 𝑏𝑘 = 𝑏𝑘−1 + 𝑛𝑘−1, in which 𝑛𝑘−1 is virtual noise of the current sensor
bias used only for the calculation. The output equation of the model describing the relationship of the cell
voltage, the current, SoC, the hysteresis, and the diffused resistor currents is written as (4).
𝑦𝑘 = 𝑂𝐶𝑉(𝑧𝑘) + 𝑀ℎ𝑘 − 𝑅1𝑖𝑅1,𝑘 − 𝑅2𝑖𝑅2,𝑘 − 𝑅2(𝑖𝑘 − 𝑏𝑘) (4)
The state vector related to the diffused resistor currents, the hysteresis, SoC and the current sensor bias as is
formed as the (5).
x𝑘 = [𝑖𝑅1,𝑘+1 𝑖𝑅2,𝑘+1 ℎ𝑘+1 𝑧𝑘+1 𝑏𝑘+1]𝑇
(5)
Define the system matrices and output matrix of the model as.
A𝑅𝐶 = [
𝑒𝑥𝑝 (−
∆𝑡
𝑅1𝐶1
) 0
0 𝑒𝑥𝑝 (−
∆𝑡
𝑅2𝐶2
)
] , B𝑅𝐶 = [
1 − 𝑒𝑥𝑝 (−
∆𝑡
𝑅1𝐶1
)
1 − 𝑒𝑥𝑝 (−
∆𝑡
𝑅2𝐶2
)
] , A𝐻,𝑘 = 𝑒𝑥𝑝 (|
𝑖𝑘𝜂𝑘∆𝑡
𝑄
|)

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1338
A(𝑖𝑘) = [
1 0 0
0 A𝑅𝐶 0
0 0 A𝐻,𝑘
] , B(𝑖𝑘) = [
−
𝜂𝑘∆𝑡
𝑄
0
B𝑅𝐶 0
0 1 − A𝐻,𝑘
] , C𝑘 = [0 −𝑅1 −𝑅2 𝑀], D𝑘 = −𝑅0
Suppose the input and output vectors of the model as (6).
u𝑘 = [
𝑖𝑘 + 𝑏𝑘
𝑠𝑖𝑔𝑛(𝑖𝑘 + 𝑏𝑘)
] , 𝑦𝑘 = 𝑣𝑘 (6)
Finally, the SECM of the cell is written as (7).
{
x𝑘+1 = A(𝑖𝑘)x𝑘 + B(𝑖𝑘)u𝑘
𝑦𝑘 = 𝑂𝐶𝑉( 𝑧𝑘, 𝑇𝑘) + Cx𝑘 + Du𝑘
(7)
Model (7) is used to estimate the SoC for the cell with input u𝑘 and output 𝑦𝑘 without measurement
noise. When measuring u𝑘 and 𝑦𝑘, the two measurement noises 𝑛𝑖,𝑘 and 𝑛𝑣,𝑘 were mixed in the curent and
voltage as described in Figure 3(a). So that, the input vector and output vector of the model is written as (8).
u𝑛,𝑘 = [
𝑖𝑘 + 𝑏𝑘 + 𝑛𝑖,𝑘
𝑠𝑖𝑔𝑛(𝑖𝑘 + 𝑏𝑘+𝑛𝑖,𝑘)
] , 𝑦𝑛,𝑘 = 𝑣𝑘 +𝑛𝑣,𝑘 (8)
With the input and output vectors in the (8), the model (7) is rewritten as (9).
{
x𝑛,𝑘+1 = A(𝑖𝑛,𝑘)x𝑛,𝑘 + B(𝑖𝑛,𝑘)u𝑛,𝑘
𝑦𝑛,𝑘 = 𝑂𝐶𝑉( 𝑧𝑛,𝑘, 𝑇𝑘) + Cx𝑛,𝑘 + Du𝑛,𝑘
(9)
In which x𝑛,𝑘+1 is state vector affected by measurement noise formed as (10).
x𝑛,𝑘 = [𝑖𝑅1𝑛,𝑘+1 𝑖𝑅2𝑛,𝑘+1 ℎ𝑛,𝑘+1 𝑧𝑛,𝑘+1 𝑏𝑛,𝑘+1]𝑇
(10)
Suppose the error w𝑘 = x𝑘+1 x𝑛,𝑘+1 to be the system noise of the model, 𝜉𝑘 = 𝑦𝑘−𝑦𝑛,𝑘 is the
output noise of the model, we have (11).
{
x𝑘+1 = A(𝑖𝑛,𝑘)x𝑛,𝑘 + B(𝑖𝑛,𝑘)u𝑛,𝑘 + w𝑘
𝑦𝑘 = 𝑂𝐶𝑉( 𝑧𝑘, 𝑇𝑘) + Cx𝑘 + Du𝑛,𝑘 + 𝜉𝑘
(11)
The model (11) is used to estimate the state vector x𝑘+1, in that vector there are the SoC 𝑧𝑘 and the current
sensor bias 𝑏𝑘 need to be estimated. In this work, we use LIB Samsung ICR18650-22P [20] with parameters
𝑄 = 2200 𝑚𝐴ℎ, 𝑉
𝑚𝑎𝑥 = 4.25 𝑉, 𝑉𝑚𝑖𝑛 = 2.75 𝑉, nominal voltage 3.62 𝑉. The test temperature range is
considered from −5 °𝐶 to 45 °𝐶. The values of model’s parameters determined by practical tests are given in
Table 1.
Table 1. The cell’s model parameters of LIB Samsung ICR18650-22P
T0
C -5 0
C 5 0
C 15 0
C 25 0
C 35 0
C 45 0
C
𝐤
1.0869 0.9803 1.0220 0.95 1.0542 1.0399
𝐐 (𝐀𝐡) 2.1596 2.1877 2.1943 2.175 2.1515 2.1523
 243.1946 78.4915 63.6762 250.0074 170.6407 151.3064
𝐌𝟎 (𝐕) 0.0072 0.0049 0.0035 0.002 0.0036 0.0024
𝐌 (𝐕) 0.0347 0.0257 0.0218 0.02 0.0201 0.0185
𝐑𝟎 (Ω) 0.013 0.023 0.0292 0.033 0.0012 0.0011
𝐑𝟏𝐂𝟏 (𝐬) 0.6124 1.7555 0.3227 0.899 0.6997 0.4630
𝐑𝟐𝐂𝟐 (𝐬) 3.9035 7.5994 8.1118 10.023 8.1840 6.5319
𝐑𝟏 (Ω) 0.0204 0.0203 0.0201 0.019 0.019 0.0019
𝐑𝟐 (Ω) 0.0494 0.0376 0.0288 0.022 0.0136 0.0134

6. Int J Elec & Comp Eng ISSN: 2088-8708 
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3. PROPOSED SOC ESTIMATION ALGORITHM FOR 𝑵 CELLS CONNECTED IN SERIES
USING TWO SPKFS
Suppose that there is 𝑁 cells LIB connected in series as shown in Figure 5, the problem is to
estimate SoCs of 𝑁 cells with the input and output signals affected by noises as (8), the noise distributions
are considered as Gaussian-distributed. SPKF is the filter with high precision aimed at nonlinear system, in
practice, the SPKF has some advantages compared to the EKF. This filter allows performance KF for
nonlinear systems without the linearization steps. The set of weighted samples, called sigma points, is used
for normalizing mean and covariance of a probability distribution. The sigma points of input distributions are
mapped thought the nonlinear dynamic model functions to get the sigma points of output distribution
[21], [22]. SPKF allows estimation states of nonlinear dynamic systems more accurately than the EKF does,
but the computational complexity is same as EKF [23]–[25]. SPKF is easily implemented because there is no
requirement of the Jacobi matrix calculation. The accuracy of state estimation using the SPKF mainly
depends on the strategies of choosing sigma-points [26], [27]. Normally, we need to estimated SoC 𝑁 times
for 𝑁 cells. If the number of cells 𝑁 is large, there is a large amount of calculation [18], [19]. To reduce the
SoC estimation times, in this work we use only two SPKFs, the first filter estimates the mean of 𝑁 SoC and
the current sensor bias, the second filter estimates 𝑁 SoC differences from the average SoC, the estimated
SoC of each cell then is calculated by the sum of the average SoC determined by the first filter and the
difference SoC estimated by the second filter. In one sampling time interval, the SoC estimation algorithm is
implemented two times instead of 𝑁 times.
Figure 5. 𝑁 LIB cells connected in series
Consider the second model of cell in the (11) with input and output in the (8). The SOC estimation
algorithm flowchart for 𝑁 cell connected in series is conducted as:
The SOC estimation algorithm flowchart
Begin
Initialization
For each sampling time 𝑘, 𝑘 = 1,2,3, … , ∞, do
Measure the current of cell 𝑖𝑛,𝑘 ; Measure voltages of 𝑁 cells 𝑦𝑛,𝑘
1
, 𝑦𝑛,𝑘
2
, … , 𝑦𝑛,𝑘
𝑁
;
Measure ambient temperature of cell 𝑇𝑘
Estimate state vector x
̂𝑘 = [𝑖̂𝑅1,𝑘+1 𝑖̂𝑅2,𝑘+1 ℎ
̂𝑘+1 𝑧̂𝑘+1 𝑏
̂𝑘+1]
𝑇
by first SPKF
For each cell 𝑖 = 1,2,3, … , 𝑁 do
Estimate the SOC difference ∆𝑧̂𝑘
(𝑖)
by second SPKF; Estimate
the SoC for cell 𝑧̂𝑘
(𝑖)
= 𝑧̂𝑘 + ∆𝑧̂𝑘
(𝑖)
End
End
End
3.1. Algorithm to estimate the average SoC for the N cells by the first SPKF filter
The first filter estimates the mean of 𝑁 SoC and the current sensor bias as steps following:
− Form the augmented state vector at sampling time 𝑘 defined by x
̂𝑘−1
𝑎+
, including the estimated state
vector at the sampling time 𝑘 − 1 the mean vector of the system noise w
̅ and mesurement noise ξ̅:
x
̂𝑘−1
𝑎,+
= [x
̂𝑘−1
+
w
̅ ξ̅] (12)
− Form the augmented covariance matrix (CM) of state estimation error vector by using CM of state
estimation error at sampling time 𝑘 − 1, the CM of system noises and the CM of measurement noises:

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σx
̃,𝑘−1
𝑎,+
= 𝑑𝑖𝑎𝑔[σx
̃,𝑘−1
+
σw σ𝜉] (13)
− Set up matrix of 𝑝 + 1 sigma points:
X𝑘−1
𝑎,+
= {x
̂𝑘−1
𝑎,+
, x
̂𝑘−1
𝑎,+
+ 𝛾√σx
̃,𝑘−1
𝑎,+
, x
̂𝑘−1
𝑎,+
− 𝛾√σx
̃,𝑘−1
𝑎,+
, } ≜ [X𝑘−1
x,+
X𝑘−1
𝑤,+
X𝑘−1
ξ,+
]
𝑇
(14)
− Calculate sigma points matrix at sampling time 𝑘 by using the cell model’s dynamic function in (11):
X𝑘,𝑗
x,−
= A𝑘(𝑖𝑛,𝑘)X𝑘−1,𝑗
x,+
+ B𝑘(𝑖𝑛,𝑘)u𝑛,𝑘+X𝑘−1,𝑗
𝑤,+
𝑗 = 0,1,2, … 𝑝 (15)
− Estimate the priori state vector at the sampling time 𝑘:
x
̂𝑘
−
= ∑ 𝛼𝑗
𝑚
X𝑘,𝑗
x,−
𝑝
𝑗=0 (16)
− Update the CM of the state estimation errors:
σx
̃,𝑘
−
= ∑ 𝛼𝑗
𝑐
(X𝑘,𝑗
x,−
− x
̂𝑘
−
)
𝑝
𝑗=0 (X𝑘,𝑗
x,−
− x
̂𝑘
−
)
𝑇
(17)
− Calculate the output sigma points matrix at the sampling time 𝑘 by using the output equation in the
model (11):
Y𝑘 = 𝑂𝐶𝑉(𝑧̂𝑘, 𝑇𝑘) + C𝑘X𝑘,𝑗
𝑥,−
+ D𝑘u𝑛,𝑘 + X𝑘−1
ξ,+
(18)
− Estimate the output voltage of cells as (19):
y
̂k = ∑ 𝛼𝑗
𝑚
𝑝
𝑗=0 Y𝑘 (19)
− Update the CM of output voltage estimation errors:
σy,
̃𝑘
−
= ∑ 𝛼𝑗
𝑐
𝑝
𝑗=0 (Y𝑘 − y
̂k )(Y𝑘 − y
̂k )𝑇
(20)
− Update the covariance matrices of the state vector estimation error and output voltage estimation error:
σx
̃,𝑦,
̃𝑘
−
= ∑ 𝛼𝑗
𝑐
(X𝑘,𝑗
x,−
− x
̂𝑘
−
)
𝑝
𝑗=0 (Y𝑘 − y
̂k )𝑇
(21)
− Update the estimation gain matrix as (22):
L𝑘 =
σx
̃,𝑦,
̃ 𝑘
−
σy,
̃𝑘
− (22)
− Update the estimation state vector by using the estimation gain matrix, the measured real output
voltages and the estimated output voltage of cells:
x
̂𝑘
+
= x
̂𝑘
−
+ L𝑘 (y𝑘 − y
̂k ) (23)
− Update the CM of state estimation error vector:
σx
̃,𝑘
+
= σx
̃,𝑘
−
− L𝑘σy,
̃𝑘
−
L𝑘
𝑇
(24)
3.2. SoC differences estimation for 𝑵 cells by second SPKF
The second SPKF estimates the SoC differences ∆𝑧̂𝑘
(𝑖)
, 𝑖 = 1,2, … , 𝑁, for 𝑁 cells at the sampling
time 𝑘 as following steps:

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− Form the augmented state vector of cell 𝑖 at the sampling time 𝑘 − 1 by using the SoC difference ∆𝑧̂𝑘
(𝑖)
and the mean vector of the SoC difference noises 𝑤
̅𝑧∆
(𝑖)
:
∆𝑧̂𝑘−1
𝑎,(𝑖)+
= [∆𝑧̂𝑘
(𝑖)
, 𝑤
̅𝑧∆
(𝑖)
] (25)
− From the augmented CM of SoC difference estimation error of cell 𝑖, by using the CM of SoC
difference estimation error of cell 𝑖 at sampling time 𝑘 − 1 and the CM of SoC difference noise:
𝜎∆𝑧,
̃𝑘−1
𝑎,(𝑖)+
= 𝑑𝑖𝑎𝑔 (𝜎∆𝑧,
̃𝑘−1
(𝑖)+
, 𝜎𝑤∆𝑧
(𝑖) ) (26)
− Set up the matrix of 𝑝 + 1 sigma points of cell 𝑖:
∆𝜒𝑘−1
𝑎,(𝑖)+
= {∆𝑧̂𝑘−1
𝑎,(𝑖)+
, ∆𝑧̂𝑘−1
𝑎,(𝑖)+
+ 𝛾√𝜎∆𝑧
̃,𝑘−1
𝑎,(𝑖)+
, ∆𝑧̂𝑘−1
𝑎,(𝑖)+
− 𝛾√𝜎∆𝑧
̃,𝑘−1
𝑎,(𝑖)+
≜ {∆𝜒𝑘−1,𝑗
∆𝑧,(𝑖)+
, ∆𝜒𝑘−1,𝑗
𝑤∆𝑧
+
}
𝑇
(27)
− Calculate the state sigma point matrix of cell 𝑖 at the sampling time 𝑘:
∆𝜒𝑘,𝑗
∆𝑧,(𝑖)−
= ∆𝜒𝑘−1,𝑗
∆𝑧,(𝑖)+
−
((𝑖𝑛,𝑘−𝑖̂𝑏,𝑘)+∆𝜒𝑘−1,𝑦
𝑤∆𝑧
+
)
3600𝑄
𝑗 = 0,1,2, . . , 𝑝 (28)
− Estimate the SoC difference of cell 𝑖 at the sampling time:
∆𝑧̂𝑘
(𝑖)−
= ∑ 𝛼𝑗
𝑚
∆𝜒𝑘,𝑗
∆𝑧,(𝑖)
𝑝
𝑗=0 (29)
− Update the CM of SoC difference estimation error of cell:
σ∆𝑧
̃,𝑘
(𝑖)−
= ∑ 𝛼𝑗
𝑐
(∆𝜒𝑘,𝑗
∆𝑧,(𝑖)−
− ∆𝑧̂𝑘
(𝑖)−
)
𝑝
𝑗=0 (∆𝜒𝑘,𝑗
∆𝑧,(𝑖)−
− ∆𝑧̂𝑘
(𝑖)−
)
𝑇
(30)
− Calculate the output sigma points matrix of cell 𝑖:
Y𝑘
(𝑖)
= 𝑂𝐶𝑉(𝑧̅𝑘 + ∆𝑧̂𝑘
(𝑖)
, 𝑇𝑘) − 𝑖𝑛,𝑘∆𝑅(𝑖)
+
𝑀(𝑖)
ℎ
̂𝑘 − 𝑅1
(𝑖)
𝐶1
(𝑖)
𝑖̂𝑅,1 − 𝑅2
(𝑖)
𝐶2
(𝑖)
𝑖̂𝑅,2 − 𝑅0(𝑖𝑛,𝑘 − 𝑖̂𝑏
𝑘
) (31)
− Estimate the output voltage of cell 𝑖:
𝑦
̂𝑘
(𝑖)
= ∑ 𝛼𝑗
𝑚
𝑝
𝑗=0 Y𝑘
(𝑖)
(32)
− Update the CM of output voltage estimation error of cell 𝑖:
𝜎𝑦,
̃ 𝑘
(𝑖)
= ∑ 𝛼𝑗
𝑐
𝑝
𝑗=0 (Y𝑘
(𝑖)
− 𝑦
̂𝑘
(𝑖)
)(Y𝑘
(𝑖)
− 𝑦
̂𝑘
(𝑖)
)
𝑇
(33)
− Update the CM of SoC difference estimation error and output voltage estimation error of cell 𝑖:
𝜎∆𝑧
̃,𝑦
̃,𝑘
(𝑖)−
= ∑ 𝛼𝑗
𝑐
(∆𝜒𝑘,𝑗
∆𝑧,(𝑖)−
− ∆𝑧̂𝑘
(𝑖)−
)
𝑝
𝑗=0 (Y𝑘
(𝑖)
− 𝑦
̂𝑘
(𝑖)
)
𝑇
(34)
− Update the state estimation matrix of cell:
𝐿𝑘
(𝑖)
=
𝜎∆𝑧
̃,𝑦
̃,𝑘
(𝑖)−
𝜎𝑦
̃ ,𝑘
(𝑖) (35)
− Estimate the SoC difference of:

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∆𝑧̂𝑘
(𝑖)+
= ∆𝑧̂𝑘
(𝑖)−
+ 𝐿𝑘
(𝑖)
(𝑦𝑛,𝑘
(𝑖)
−𝑦
̂𝑘
(𝑖)
) (36)
− Update the CM of SoC difference estimation error of cell 𝑖:
𝜎∆𝑧
̃,𝑘
(𝑖)+
= 𝜎∆𝑧
̃,𝑘
(𝑖)−
− 𝐿𝑘
(𝑖)
𝜎𝑦,
̃ 𝑘
(𝑖)
𝐿𝑘
(𝑖)𝑇
(37)
3.3. SoC estimation of cells
The SoC of cell 𝑖 calculated as (38):
𝑧̂𝑘
(𝑖)+
= 𝑧̂𝑘
+
+ ∆𝑧̂𝑘
(𝑖)+
(38)
4. RESULTS AND DISCUSSION
The experimental system of SoC estimation is described in Figure 6, the components in the
experimental system are listed as: i) The LiB samsung is formed by 7 cells connected in series with
parameters listed in Table 1; measurement modul is created by 7 voltage difference measurement circuits, 1
current measurement curcuit, 1 temperature sensor. The voltage difference signals are to the digital signals
10 bit and transferred to the personal computer (PC) by RS485; ii) the PC implements the SoC estimation
program for cells, the configuration of PC is core (TM) i3-6100 CPU@3.70 GHZ, 4 cores, RAM 8 GB; the
MATLAB software is used to implement the SPKF algorithms, the LabVIEW 2018 [22] is used to make the
graphic user interface, the MATLAB software is embedded to the LabVIEW to run the SoC estimation;
iii) the load is motor BLDC, YONG-d67 240 W, 24 V, 10 A, the maximum speed is 295 rmp; iv) the charge
source is owon, 195 W with 2 programmable channel 36 V, maximum current 6 A; and v) the SoC estimation
algorithms are coded in MATLAB script, then embedded to the LabVIEW environment. The sampling time
interval is 1 second. The algorithm flowchart of SoC estimation for 𝑁 cell in shown in Figure 7.
Figure 6. The experimental system of SoC estimation
Figure 7. The algorithm flowchart of SoC estimation

10. Int J Elec & Comp Eng ISSN: 2088-8708 
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1343
4.1. The SoC estimation results for 1 cell
In this part, the SoC estimation results for 1 cell is presented, the Figure 8(a) describes the current,
voltage and temperature of the cell. The noise distributions of measured currrent, voltage and temperature are
plotted in Figure 8(b), the Figure 8(c) and the Figure 8(d). The current noise was distributed inside from
−0.8 𝐴 to 0.8 𝐴 caused by shunt-resistor with the sensitivity 15 𝑚𝑉/𝐴. The range of the voltage noise is
range of -0.03 𝑉 to 0.03 𝑉, the distribution of temperature noise is in the range from −0.7 0
𝐶 to 0.7 0
𝐶.
Figure 9 shows the results obtained when performing the SoC estimation algorithm corresponding to current,
temperature and voltage inputs as shown in Figure 8 using the second-order model of cell with the
parameters described in Table 1.
The Figure 9(a) describes the comparison between real cell voltage and estimated voltage calculated
by cell model, the voltage error is plotted in Figure 9(b) with 0.1 𝑉 and −0.075 𝑉 of maximum for discharge
and charge, respectively. This maximum error is caused when there is the changing of charge to discharge
vice versa. The cause of voltage error is the dependence of model on the temperature varying in the test, the
parameters of cell were determined at particular temperatures − 5 0
𝐶, 0 0
𝐶, 5 0
𝐶, 10 0
𝐶, 15 0
𝐶, 20 0
𝐶,
25 0
𝐶, 30 0
𝐶, 35 0
𝐶. In the test, the temperature of cell raises from 25.5 0
𝐶 at to 31.5 0
𝐶 as plotted in
Figure 9(c). The Figure 9(c) also plotted the estimated SoC of cell. After 1 hour with the discharging (3 𝐴)
and the charging (1.5 𝐴) alternately, The SoC of cell tends to decrease from 92.5% at beginning to 65% at
the end of the test. The SoC estimation error is shown in Figure 9(d) by comparing to criteria 𝑧̂𝑘 ± 3𝜎.
The criteria 3𝜎 decreases gradually from the beginning of the test to the end of the test as plotted in
Figure 9(e), we can see that the accuracy of SoC estimation of the cell described by the criteria 3σ at the end
of test was 0.0125%. The Figure 9(f) plots the updating of matrix 𝜎𝑥
̃
+
by the time. At the times, there are the
changings of charge to discharge vice versa, the voltage errors are large, some elements of matrix 𝜎𝑥
̃
+
need to
be changed in order to keep estimating SoC smoothly.
(a) (b)
(c) (d)
Figure 8. The measurements and distributions, (a) the current, voltage and temperature of cell in the test,
(b) the noise distributions of current, (c) temperature, and (d) voltage
Voltage (V)
Current
(A)
Temperature
(
o
C)
Voltage
(V)
Temperature
(o
C)
Current (A)

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1344
(a) (b)
(c)
(d) (e)
(f)
Figure 9. The SoC estimation results for 1 cell, (a) the voltage and estimated voltage, (b) voltage error, (c) the
estimated SoC and temperature, (d) estimated SoC±3σ, (e) criteria 3σ, , and (f) the matrix 𝛔𝐱
̃,𝑘
+
+3sigma
Voltage
(V)
Current (A)
SoChat
(%)
SoChat
-3sigma
The
matrix
𝛔
𝐱
̃
,𝑘
+
Time (s)
Voltage
(V)
SoChat
(%)
Current
(A)
Temperature (o
C)

12. Int J Elec & Comp Eng ISSN: 2088-8708 
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4.2. The SoC estimation results for 7 cells
In this part, the SoC estimation results for 7 cells connected in series are presented. Suppose that all
cells are the same with parameters listed in Table 1. The Figure 10(a) plots the cells current, the Figure 10(b)
and Figure 10(c) depicts the voltages of cells and temperature, respectively, the distributions of noises in
current, voltages and temperature are plotted in Figures 10(d), 10(e) and 10(f). The estimated SoC for 7 cells
is presented in Figure 11 and Figure 12 (in appendix), in which the Figure 11(a) is the estimated average SoC
for 7 cells, the Figure 11(b) plots the estimated SoC differences of cells.
(a) (b)
(c) (d)
(e) (f)
Figure 10. The measurements and distributions, (a) the current of 7 cells, (b) the voltages of cells, (c) the
temperature of cell, (d) the distributions of noise in current, (e) voltages and (f) temperature are plotted
The Figure 12 plots the estimated SoC of each cell and comparing with criteria 3σ, in which:
Figure 12(a) and Figure 12(b) are the estimated SoC and comparing with criteria 3σ for cell 1, similarity for
the cells from 2 to 7 are plotted in Figure 12(c) to Figure 12(o). The Figure 13(a) shows the varying of matrix
𝜎𝑧
̃
+
in the test, the Figure 13(b) plots the varying of criteria 3. The mean of the criteria 3 for cells at the
end of the test is approximately 0.5%. Figures 12 and 13 see in appendix.
Voltage of cell 1
Voltage of cell 2
Voltage of cell 3
Voltage of cell 4
Voltage of cell 5
Voltage of cell 6
Voltage of cell 7

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(a) (b)
Figure 11. The estimated SoC for 7 cells, (a) the estimated average SoC, and (b) the estimated SoC difference
of cells
5. CONCLUSION
This paper deals with the method for SoC estimation of LIB with cells connected in series. In this
works, the cell’s model is represented by the second-order equivalent circuit model taking into account the
measurement disturbances and the current sensor bias. Using 02 SPKF filters to estimate the average SoC of
cells, the current sensor bias and the SoC differences, the estimated SoC of each cell is is calculated by sum
of pack’s everage SoC estimated by first SPKF and SoC differences estimated by second SPKF. This allows
reducing the SoC estimation times to 2 times, this is a very effective solution when the battery pack has a
large number of cells. The practical test is applied to 7 samsung ICR18650-22P lithium-Ion battery cells with
the discharging (3 𝐴) and the charging (1.5 𝐴) alternately in 1 hour. The estimated SoC is quite accuracy
with the criteria 3  of estimated SoC error is approximately 0.5. In this work, we suppose that all cells are
physically identical, the equivalent circuit model with parameters listed in Table 1 are linearized at some
particular temperatures and applied for all cells. This is clear evidence that those factors contribute to the SoC
and current sensor bias estimation error observed in the test. In future work, the model of each cell to be
updated adaptively by temperature will be researched.
APPENDIX
(a) (b) (c)
(d) (e) (f)
Voltage 2 (V)
-3sigma
Estimated SoC of
cell 3
+3sigma
-3sigma
Estimated
SoC of cell 2
+3sigma
-3sigma
Estimated SoC of
cell 1
+3sigma
Volrage 3 (V)
Estimated SoC of cell 3 (%)
Temperature (°C)
Estimated
SoC of
cell 2(%)
Temperature (°C)
Voltage 1 (V)
Estimated SoC
of cell 1 (%)
Temperature (°C)
Voltage
1
(V)
Temperature (°C)
Estimated
SoC
of
cell
1
(%)
Estimated
SoC
of
cell
1
(%)
Estimated
SoC
of
cell
2
(%)
Temperature (°C)
Voltage
2
(V)
Voltage
3
(V)
Estimated
SoC
of
cell
3
(%)
Temperature (°C)

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(g) (h) (i)
(j) (k) (l)
(m) (n)
Figure 12. The estimated SoC for each cell and the comparing with criteria 3, (a) the measurements for cell
1, (b) estimated SoC for cell 1, (c) the measurements for cell 2, (d) estimated SoC for cell 2, (e) the
measurements for cell 3, and (f) estimated SoC for cell 3, (g) the measurements for cell 4, (h) estimated SoC
for cell 4, (i) the measurements for cell 5, (j) estimated SoC for cell 5, (k) the measurements for cell 6, (l)
estimated SoC for cell 6, (m) the measurements for cell 7, and (n) estimated SoC for cell 7
(a) (b)
Figure 13. The matrix σz
̃
+
and criteria 3 , (a) the varying of the matrix σz
̃
+
in the test and (b) criteria 3 
Estimated SoC of cell 7 (%)
Estimated
SoC of cell 5
+3sigma
-3sigma
Estimated
SoC of cell 6
+3sigma
-3sigma
Voltage 6 (V)
Estimated SoC of cell 6 (%)
Temperature (°C)
Voltage 4 (V)
Temperature (°C)
Voltage 5 (V)
Temperature (°C)
Estimated
SoC of cell 4
+3sigma
-3sigma
Estimated SoC of cell 4 (%) Estimated SoC of cell 5 (%)
Voltage
4
(V)
Estimated
SoC
of
cell
4
(%)
Temperature (°C)
Estimated
SoC
of
cell
5
(%)
Voltage
5
(V)
Temperature (°C)
Temperature (°C)
Estimated
SoC
of
cell
6
(%)
Voltage
6
(V) Temperature (°C)
Voltage 7 (V)
Estimated
SoC of cell 7
+3sigma
-3sigma
Temperature (°C)
Estimated
SoC
of
cell
7
(%)
Estimated
SoC
of
cell
7
(%)
Voltage
7
(V)

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ACKNOWLEDGEMENTS
This research was supported by Thai Nguyen University of Technology, TNUT, Vietnam.
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16. Int J Elec & Comp Eng ISSN: 2088-8708 
State of charge estimation for lithium-ion batteries connected in series using … (Chi Nguyen Van)
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BIOGRAPHIES OF AUTHORS
Chi Nguyen Van received the PhD degree in Instrument and Control Engineering
from Hanoi University of Science and Technology (HUST) in 2012. He is currently working
as a Associate Professor with Institute of High-Technology Research and Development for
Industry (RIAT), Thai Nguyen University of Technology (TNUT), Viet Nam. His research
interests include nonlinear control, adaptive control, optimal control and system identification
for Lithium Ion Batterry. He can be contacted at email: ngchi@tnut.edu.vn
Thuy Nguyen Vinh received the MSs degree in Automation Engineering from
Thai Nguyen University of Technology (TNUT) in 2005. He is currently working as a
researcher with Institute of High-Technology Research and Development for Industry (RIAT),
Thai Nguyen University of Technology (TNUT) His research interests optimal control and
system identification for Lithium Ion Batterry. He can be contacted at email: nguyenvinhthuy-
tdh@tnut.edu.vn